91Ó°ÊÓ

Parametric equations are a powerful tool in mathematics for describing curves, surfaces, and more. Rather than expressing relationships using standard Cartesian coordinates, parametric equations introduce one or more parameters to define a set of equations. In the context of surfaces of revolution, parametrics are used to describe how a surface behaves as it revolves around an axis. When manipulating these equations, each parameter plays a specific role:

• The parameter \( z \) represents the height along the \( z \)-axis, which is the axis of revolution in this exercise.
• Another parameter, usually an angular measure, like \( \theta \), indicates the amount of rotation around the axis. In our example, \( \theta \) ranges from \( 0 \) to \( 2\pi \), enabling a full revolution.

In our specific problem, the parametric representation of the surface is derived from the function \( x = e^z \) by introducing \( \theta \) for the angle of revolution. This leads to the equations: \[ \begin{align*}x &= e^z \cos(\theta), \y &= e^z \sin(\theta), \z &= z.\end{align*} \] These equations depict every point on the surface as \( z \) varies from \( 0 \) to \( 1 \) and \( \theta \) spans a complete circle.

Polar coordinates offer an alternative to Cartesian coordinates, particularly useful for circular or rotational systems. They define a point based on its distance from a fixed central point (the pole) and the angle from a reference direction. Understanding these basics of polar systems is crucial when dealing with surfaces of revolution like this one:

• The radial coordinate \( r \) signifies the distance from the axis of revolution, which is derived from the original function \( r = e^z \).
• The angular coordinate \( \theta \) expresses the angle of revolution about the \( z \)-axis, reflecting full rotation over \( 0 \leq \theta < 2\pi \).

To convert a function defined in this manner to a three-dimensional object when revolving around the \( z \)-axis, we use: \[ x = r \cos(\theta) \y = r \sin(\theta) \z = z \]This setup allows us to graphically illustrate the surface in terms of its radial and angular positions as it rotates around the \( z \)-axis.

The concept of graph revolving involves rotating a graph about an axis to form a three-dimensional surface. In this exercise, you've revolved the graph of the function \( x = e^z \) about the \( z \)-axis. To visualize graph revolving:

• Imagine the line \( x = e^z \) extending vertically as \( z \) increases. As \( z \) moves from 0 to 1, the graph forms a curve in the \( xz \)-plane.
• When this curve revolves around the \( z \)-axis, it sweeps through the surrounding 3D space, creating a surface.
• This surface is represented through the parametrization that incorporates both the function definition (\( e^z \)) and a rotational parameter (\( \theta \)).

By using parametric equations with polar coordinates, \[ (x, y, z) = (e^z \cos(\theta), e^z \sin(\theta), z) \]you've expressed the surface of revolution in a form that is both mathematically precise and visually interpretable.